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\begin{table}[!tbp]
\caption{Simulation results: Linear outcome models\label{tb_Linear_2}} 
{\centering
\begin{tabular}{lrrcrcrrcrrcrrcrcrr}
\hline
\multicolumn{1}{l}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}\tabularnewline
\cline{2-3} \cline{7-8} \cline{13-14} \cline{18-19}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Linear outcome model 1: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ -9.78$&$10.39$&&$$&&$ -9.97$&$ 10.09$&&$$&$$&&$10.05$&$10.68$&&$$&&$ 9.95$&$10.07$\tabularnewline
~~\textbf{nDBW DR}&$  0.54$&$ 2.89$&&$$&&$  0.17$&$  1.31$&&$$&$$&&$ 0.74$&$ 2.80$&&$$&&$ 0.09$&$ 1.20$\tabularnewline
~~MLE DR&$  0.36$&$ 3.65$&&$$&&$  0.01$&$  1.74$&&$$&$$&&$ 0.44$&$ 3.74$&&$$&&$ 0.00$&$ 1.76$\tabularnewline
~~CBPS DR&$  0.49$&$ 3.25$&&$$&&$  0.07$&$  1.59$&&$$&$$&&$ 0.38$&$ 3.08$&&$$&&$ 0.02$&$ 1.46$\tabularnewline
~~Calibrated weighting DR&$  0.54$&$ 2.97$&&$$&&$  0.12$&$  1.41$&&$$&$$&&$ 0.42$&$ 2.74$&&$$&&$ 0.04$&$ 1.22$\tabularnewline
~~Entropy balancing DR&$  1.15$&$ 3.16$&&$$&&$  0.91$&$  1.64$&&$$&$$&&$ 1.40$&$ 3.10$&&$$&&$ 1.11$&$ 1.66$\tabularnewline
~~True propensity score DR~~&$  0.37$&$ 3.57$&&$$&&$  0.04$&$  1.81$&&$$&$$&&$ 0.62$&$ 4.08$&&$$&&$ 0.05$&$ 1.96$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Linear outcome model 1: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$-10.02$&$10.64$&&$$&&$ -9.97$&$ 10.10$&&$$&$$&&$ 9.92$&$10.51$&&$$&&$10.01$&$10.14$\tabularnewline
~~\textbf{nDBW DR}&$ -1.98$&$ 3.73$&&$$&&$ -2.61$&$  2.99$&&$$&$$&&$ 2.49$&$ 3.81$&&$$&&$ 1.79$&$ 2.18$\tabularnewline
~~MLE DR&$ -5.97$&$22.52$&&$$&&$-16.30$&$129.23$&&$$&$$&&$ 3.11$&$ 4.49$&&$$&&$ 3.07$&$ 3.39$\tabularnewline
~~CBPS DR/BRDR&$ -2.73$&$ 4.36$&&$$&&$ -3.57$&$  3.97$&&$$&$$&&$ 3.07$&$ 4.39$&&$$&&$ 3.33$&$ 3.63$\tabularnewline
~~Calibrated weighting DR&$ -2.14$&$ 3.83$&&$$&&$ -2.76$&$  3.14$&&$$&$$&&$ 2.31$&$ 3.70$&&$$&&$ 2.23$&$ 2.58$\tabularnewline
~~Entropy balancing DR&$ -1.52$&$ 3.56$&&$$&&$ -1.95$&$  2.46$&&$$&$$&&$ 3.79$&$ 4.80$&&$$&&$ 3.77$&$ 4.00$\tabularnewline
~~True propensity score DR~~&$  0.31$&$ 3.66$&&$$&&$  0.09$&$  1.77$&&$$&$$&&$ 0.47$&$ 4.09$&&$$&&$ 0.12$&$ 1.97$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Linear outcome model 2: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ -3.59$&$ 6.18$&&$$&&$ -3.76$&$  4.37$&&$$&$$&&$ 3.76$&$ 6.29$&&$$&&$ 3.73$&$ 4.34$\tabularnewline
~~\textbf{nDBW DR}&$  0.07$&$ 3.64$&&$$&&$  0.03$&$  1.65$&&$$&$$&&$ 0.62$&$ 3.66$&&$$&&$ 0.08$&$ 1.63$\tabularnewline
~~MLE DR&$ -0.06$&$ 4.89$&&$$&&$ -0.13$&$  2.19$&&$$&$$&&$ 0.43$&$ 5.14$&&$$&&$ 0.02$&$ 2.35$\tabularnewline
~~CBPS DR&$  0.06$&$ 4.40$&&$$&&$ -0.08$&$  2.01$&&$$&$$&&$ 0.34$&$ 4.37$&&$$&&$ 0.04$&$ 2.01$\tabularnewline
~~Calibrated weighting DR&$  0.10$&$ 3.77$&&$$&&$ -0.05$&$  1.72$&&$$&$$&&$ 0.37$&$ 3.73$&&$$&&$ 0.06$&$ 1.67$\tabularnewline
~~Entropy balancing DR&$  0.02$&$ 3.85$&&$$&&$ -0.08$&$  1.74$&&$$&$$&&$ 1.18$&$ 3.91$&&$$&&$ 0.95$&$ 1.91$\tabularnewline
~~True propensity score DR~~&$ -0.12$&$ 5.19$&&$$&&$ -0.09$&$  2.48$&&$$&$$&&$ 0.63$&$ 5.56$&&$$&&$ 0.07$&$ 2.62$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Linear outcome model 2: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ -3.88$&$ 6.33$&&$$&&$ -3.74$&$  4.34$&&$$&$$&&$ 3.69$&$ 6.13$&&$$&&$ 3.73$&$ 4.36$\tabularnewline
~~\textbf{nDBW DR}&$ -3.36$&$ 5.48$&&$$&&$ -3.56$&$  4.02$&&$$&$$&&$ 3.27$&$ 5.32$&&$$&&$ 2.58$&$ 3.17$\tabularnewline
~~MLE DR&$ -6.05$&$17.36$&&$$&&$ -9.29$&$ 62.15$&&$$&$$&&$ 4.34$&$ 6.43$&&$$&&$ 4.25$&$ 4.73$\tabularnewline
~~CBPS DR/BRDR&$ -4.42$&$ 6.42$&&$$&&$ -4.53$&$  4.99$&&$$&$$&&$ 4.44$&$ 6.44$&&$$&&$ 4.49$&$ 4.95$\tabularnewline
~~Calibrated weighting DR&$ -3.55$&$ 5.59$&&$$&&$ -3.73$&$  4.18$&&$$&$$&&$ 3.27$&$ 5.33$&&$$&&$ 3.15$&$ 3.66$\tabularnewline
~~Entropy balancing DR&$ -3.49$&$ 5.70$&&$$&&$ -3.51$&$  4.03$&&$$&$$&&$ 4.54$&$ 6.33$&&$$&&$ 4.45$&$ 4.86$\tabularnewline
~~True propensity score DR~~&$ -0.32$&$ 5.28$&&$$&&$ -0.02$&$  2.42$&&$$&$$&&$ 0.47$&$ 5.75$&&$$&&$ 0.07$&$ 2.63$\tabularnewline
\hline
\end{tabular}}
\parbox{0.99\textwidth}
	{Notes: This simulation compares the performance of various methods 
	for estimating propensity scores and (inverse probability) weights 
	by investigating combinations of two versions of the true outcome model 
	(Linear~1 and 2)
	and two versions of coefficients for the propensity score model (type~A and B)
	with the two different numbers of observations ($n = 200$ and $n = 1000$).
	For each estimation method, I use two propensity score model specifications 
	(correct and misspecified) and report the bias and RMSE for each in the table.}\end{table}
